Sufficient conditions for the existence of periodic solutions to some second order differential equations with a deviating argument

AbstractBy means of Mawhin's continuation theorem, we study some second order differential equations with a deviating argument: x″(t)=f(t,x(t),x(t−τ(t)),x′(t))+e(t). Some new results on the existence of periodic solutions are obtained. The interest is that we allow the degree with respect to the variables x0,x1,x2 of f(t,x0,x1,x2) to be greater than 1; and also the result (Theorem 3.2) is related to the deviating argument τ(t). Meanwhile, we give an example to demonstrate our result

Positive solutions for (n−1,1) three-point boundary value problems with coefficient that changes sign

AbstractIn this paper, we establish existence results for positive solutions for the (n−1,1) three-point boundary value problems consisting of the equation u(n)+λa(t)fu(t)=0,t∈(0,1), with one of the following boundary value conditions: u(0)=αu(η),u(1)=βu(η),u(i)(0)=0fori=1,2,…,n−2, and u(n−2)(0)=αu(n−2)(η),u(n−2)(1)=βu(n−2)(η),u(i)(0)=0fori=0,1,…,n−3, where η∈(0,1), α⩾0, β⩾0, and a:(0,1)→R may change sign and R=(−∞,+∞). f(0)>0, λ>0 is a parameter. Our approach is based on the Leray–Schauder degree theory. This paper is motivated by Eloe and Henderson (Nonlinear Anal. 28 (1997) 1669–1680)

Positive solutions of singular four-point boundary value problem with pp-Laplacian

In this paper, we deal with the following singular four-point boundary value problem with $p$-Laplacian $$\left\{\begin{aligned} &(\phi_{p}(u'(t)))'+q(t)f(t,u(t))=0,\ t\in(0,1),\\ &u(0)-\alpha u'(\xi)=0,\ u(1)+\beta u'(\eta)=0, \end{aligned}\right.$$ where $f(t,u)$ may be singular at $u=0$ and $q(t)$ may be singular at $t=0$ or $1$. By imposing some suitable conditions on the nonlinear term $f$, existence results of at least two positive solutions are obtained. The proof is based upon theory of Leray-Schauder degree and Krasnosel'skii's fixed point theorem

Solvability for second-order nonlocal boundary value problems with a p-Laplacian at resonance on a half-line

This paper investigates the solvability of the second-order boundary value problems with the one-dimensional $p$-Laplacian at resonance on a half-line $$\left\{\begin{array}{llll} (c(t)\phi_{p}(x'(t)))'=f(t,x(t),x'(t)),~~~~0<t<\infty,\\ x(0)=\sum\limits_{i=1}^{n}\mu_ix(\xi_{i}), ~~\lim\limits_{t\rightarrow +\infty}c(t)\phi_{p}(x'(t))=0 \end{array}\right.$$ and $$\left\{\begin{array}{llll} (c(t)\phi_{p}(x'(t)))'+g(t)h(t,x(t),x'(t))=0,~~~~0<t<\infty,\\ x(0)=\int_{0}^{\infty}g(s)x(s)ds,~~\lim\limits_{t\rightarrow +\infty}c(t)\phi_{p}(x'(t))=0 \end{array}\right.$$ with multi-point and integral boundary conditions, respectively, where $\phi_{p}(s)=|s|^{p-2}s$, $p>1$. The arguments are based upon an extension of Mawhin's continuation theorem due to Ge. And examples are given to illustrate our results

Multiple positive solutions for boundary value problems of second order delay differential equations with one-dimensional p-Laplacian

AbstractWe consider the boundary value problems: (ϕp(x′(t)))′+q(t)f(t,x(t),x(t−1),x′(t))=0, ϕp(s)=|s|p−2s, p>1, t∈(0,1), subject to some boundary conditions. By using a generalization of the Leggett–Williams fixed-point theorem due to Avery and Peterson, we provide sufficient conditions for the existence of at least three positive solutions to the above problems

On a class of second-order impulsive boundary value problem at resonance

We consider the following impulsive boundary value problem, x″(t)=f(t,x,x′), t∈J\{t1,t2,…,tk}, Δx(ti)=Ii(x(ti),x′(ti)), Δx′(ti)=Ji(x(ti),x′(ti)), i=1,2,…,k, x(0)=(0), x′(1)=∑j=1m−2αjx′(ηj). By using the coincidence degree theory, a general theorem concerning the problem is given. Moreover, we get a concrete existence result which can be applied more conveniently than recent results. Our results extend some work concerning the usual m-point boundary value problem at resonance without impulses

Positive solutions for a multi-point eigenvalue problem involving the one dimensional pp-Laplacian

A multi-point boundary value problem involving the one dimensional $p$-Laplacian and depending on a parameter is studied in this paper and existence of positive solutions is established by means of a fixed point theorem for operators defined on Banach spaces with cones

Infinitely many solutions of a second-order pp-Laplacian problem with impulsive condition

summary:Using the critical point theory and the method of lower and upper solutions, we present a new approach to obtain the existence of solutions to a $p$-Laplacian impulsive problem. As applications, we get unbounded sequences of solutions and sequences of arbitrarily small positive solutions of the $p$-Laplacian impulsive problem